Generalized and weighted Strichartz estimates
Jin-Cheng Jiang Chengbo Wang Xin Yu
In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension $2$ and $3$.
keywords: Generalized Strichartz estimates weighted Strichartz estimates Strauss conjecture semilinear wave equations. angular regularity
On one dimensional quantum Zakharov system
Jin-Cheng Jiang Chi-Kun Lin Shuanglin Shao
In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system (1a)-(1b) with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well-posedness in low regularity spaces (see Theorem 1.1 and Figure 1). Especially, the low regularity result for $k$ satisfies $-3/4 < k \leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for ``Schrödinger" waves.
keywords: one dimensional Quantum Zakharov system Cauchy problem low regularity. local well-posedness

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